Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data

TL;DR

The paper tackles graph topology inference for Gaussian Graphical Models when the observed mean is time-varying, exposing bias in standard Graphical Lasso (GL) under a nonzero, evolving mean. It introduces GL-ATAIS, an alternating Bayesian–frequentist framework that uses Adaptive Importance Sampling to estimate a time-varying mean $\mathbf f_r(\boldsymbol{\phi})$ via $\boldsymbol{\phi}$ and then applies GL to the centered data to estimate the sparse precision matrix $\boldsymbol{\Theta}$. The approach iteratively refines both estimates, enabling uncertainty quantification for the mean parameters and yielding substantially improved graph recovery compared to standard GL and to ATAIS alone. Numerical results on synthetic networks show that GL-ATAIS can approach the performance of an oracle with known mean, highlighting the practical impact of accounting for non-stationary means in graph learning.

Abstract

This work addresses the problem of graph learning from data following a Gaussian Graphical Model (GGM) with a time-varying mean. Graphical Lasso (GL), the standard method for estimating sparse precision matrices, assumes that the observed data follows a zero-mean Gaussian distribution. However, this assumption is often violated in real-world scenarios where the mean evolves over time due to external influences, trends, or regime shifts. When the mean is not properly accounted for, applying GL directly can lead to estimating a biased precision matrix, hence hindering the graph learning task. To overcome this limitation, we propose Graphical Lasso with Adaptive Targeted Adaptive Importance Sampling (GL-ATAIS), an iterative method that jointly estimates the time-varying mean and the precision matrix. Our approach integrates Bayesian inference with frequentist estimation, leveraging importance sampling to obtain an estimate of the mean while using a regularized maximum likelihood estimator to infer the precision matrix. By iteratively refining both estimates, GL-ATAIS mitigates the bias introduced by time-varying means, leading to more accurate graph recovery. Our numerical evaluation demonstrates the impact of properly accounting for time-dependent means and highlights the advantages of GL-ATAIS over standard GL in recovering the true graph structure.

Figures (2)

• Figure 1: Average F1-score over 100 repetitions for four algorithms, when the number of observations varies in {50, 100, 150, 200}. The ATAIS repetitions are run for T=30 iterations, with N=30000 particles.
• Figure 2: Average F1-score over 100 repetitions for four algorithms versus the number of particles in ATAIS.